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ME444: Module 3
Velocities, Jacobians and StaticForces
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Differentiation of position vectors
ttQttQQ
dtdV
BB
t
BQB
)()(lim
0
We are calculating the derivative of Qrelative to frame B.
Derivative of a vector:
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Differentiation of position vectors
Qdtd
VB
A
Q
BA
A velocity vector may be described interms of any frame:
.QBA
BQ
BA
VRV
We may write it:
VV CORGUC
Special case: Velocity of the origin of a frame relative tosome understood universe reference frame
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Example
A fi xed universal f rame
30 mph
100 mph
.70)(
.100)100()(
.
30
1
1
XRRVRRVRV
XRXRRvvV
XvVPdt
d
U
T
U
CCORG
TU
T
C
UCORG
TC
TCORG
TC
U
C
C
UT
C
UT
C
TORG
UC
CCORG
U
CORG
U
U
Both vehiclesare heeding in Xdirection of U
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Angular velocity vector:
Linear velocity attribute of a point
Angular velocity attribute of a body
Since we always attach a frame
to a body we can considerangular velocity as describingrational motion of a frame.
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Angular velocity vector:
describes the rotation of frame {B} relative to {A}
B
A
B
AMagni tude of
indicates speed of rotation
B
A
C
U
C
I n the case which there is an
understood reference frame:
direction of
indicates instantaneous axis
of rotation
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Linear velocity of a rigid body
.QBA
BBORG
A
Q
A
VRVV
We wish to describe motion
of {B} relative to frame {A}
RA
BI f rotation
is not changing with time: 7
0RAB
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Rotational velocity of a rigid body
Two frames with coincident origins
The orientation of B withrespect to A is changing
in time.
Lets consider that vectorQ is constant as viewed
from B.
B
A
QB
{A}{B}
0QBV
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Is perpendicularto and
Rotational velocity of a rigid body
QV
tQ
A
B
A
Q
A
B
AA
Q
)|)(|sin|(|||
B
A
Magnitude of differentialchange is:
QA
|| Q
Vector cross product
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Rotational velocity of a rigid body
.QRVRVBA
BB
A
Q
BA
BQ
A
In general case: QVV ABA
Q
BA
Q
A )(
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.QRVRVV
BA
BB
A
Q
BA
BBORG
A
Q
A
Simultaneous linear and rotationalvelocity
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Motion of the Links of a Robot
At any instant, each l ink of a robot in motion has some linear and
angular velocity.
Written in frame i
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Velocity of a Link
Remember that linear velocity is
associated with a point and angular
velocity is associated with a body. Thus:
The velocity of a link means the linear
velocity of the origin of the link frameand the rotational velocity of the link
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Velocity Propagation From Link to Link
We can compute the velocities of each
link in order starting from the base.
The velocity of linki+1 will be that of linki, plus whatever new velocity component
added by joint i+1.
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Rotational Velocity
Rotational velocities may be added when
both w vectors are written with respect to
the same frame.Therefore the angular velocity of linki+1
is the same as that of linkiplus a new
component caused by rotational velocityat joint i+1.
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Velocity Vectors of Neighboring Links
.1
1
111
i
i
i
i
ii
i
i
i ZR 16
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By premultiplying both sides of previousequation to:
.
.
1
1
1
1
1
1
1
1
11
11
1
1
i
i
ii
ii
ii
i
i
i
i
i
i
i
ii
ii
ii
ii
i
ZR
ZRRRR
Velocity Propagation From Link to Link
1
1
1
1
1 0
0
i
i
i
i
i Z
Note that:
Ri i1
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Linear Velocity
The linear velocity of the origin of frame
{i+1} is the same as that of the origin of
frame {i} plus a new component causedby rotational velocity of linki.
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).(
).(
1
1
1
1
1
1
1
1
i
i
i
i
i
ii
ii
i
i
i
i
i
i
ii
ii
ii
i
PvRv
PvRvR
Linear Velocity
.QRVRVV BABBA
Q
BA
BBORG
A
Q
A
Simultaneous linear and rotationalvelocity:
.11 ii
i
i
i
i
i
i Pvv By premultiplying both sides of previousequation to: Ri i
1
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.)(,
1
1
11
1
1
1
1
1
1
i
i
ii
i
i
i
i
ii
ii
i
i
ii
ii
i
ZdPvRvR
For the case that joint i+1 is prismatic:
Prismatic Joints Link
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Velocity Propagation From Link to Link
Applying those previous equations
successfully from link to link, we can
compute the rotational and linearvelocities of the last link.
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Example
A 2-li nk manipulator with rotational joints
Calculate the velocityof the tip of the armas a function of joint
rates?
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Example
Frame assignments for the two linkmanipulator
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.
1000
0100
0010001
,
1000
0100
000
,
1000
0100
0000 2
2
3
22
122
1
2
11
11
0
1
l
Tcslsc
Tcssc
T
Example
We compute link transformations:
. 11
1
1
1
1
i
i
ii
ii
ii
i ZR
.)(1
1
11
1
1
1
i
i
ii
i
i
i
i
ii
ii
i ZdPvRv 24
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.0
0
)(,
,
00
0
100
0
0
,0
0
,
0
0
0
,0
0
2
1
2221
21
212121
121
3
3
2
2
3
3
121
121
1122
22
2
2
21
2
2
1
1
1
1
1
llcl
sllcl
sl
v
cl
sl
lcs
sc
v
v
Example
Link to link transformation
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.
0
)()(
.
100
0
0
2
1
12212211
12212211
21122111
21122111
3
30
33
0
1212
1212
2
3
1
2
0
1
0
3
clclcl
slslslclclslsl
vRv
cs
sc
RRRR
Example
Velocities with respect to nonmoving base
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Derivative of a Vector Function
If we have a vector function r whichrepresents a particles position as a
function of time t:
dt
dr
dt
dr
dt
dr
dt
d
rrr
zyx
zyx
r
r
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Vector Derivatives
Weve seen how to take a derivative ofa vector vs. A scalar
What about the derivative of a vectorvs. A vector?
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Jacobian
A Jacobian is a vector derivative with respectto another vector
If we have f(x), the Jacobian is a matrix ofpartial derivatives- one partial derivative foreach combination of components of thevectors
The Jacobian is usually written as j(f,x), butyou can really just think of it as df/dx
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Jacobian
N
MM
N
x
f
x
f
x
f
x
f
x
f
x
f
x
f
J
......
............
......
...
,
1
2
2
1
2
1
2
1
1
1
xf
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Partial Derivatives
The use of the symbol instead of d forpartial derivatives just implies that it is
a single component in a vectorderivative.
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.)(
.)(
0
0
0
0
Jv
XXJY
V
In the field of robotics, we generally speak ofJacobians which relatejoint velocities toCartesian velocities of the tip of the arm.
Jacobian
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Jacobian
For a 6 joint robot the Jacobian is 6x6, .is a 6x1 and v is 6x1.
The number of rows in Jacobian is equal
to number of degrees of freedom in
Cartesian space and the number of
columns is equal to the number of joints.
)(00 JV
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Jacobian
.
0
)(
)(
2
1
12212211
1221221121122111
21122111
330
330
clclclslslslclcl
slsl
vRv
In the earlier example we had:
12212211
122122110
)( clclcl
slslsl
J
Thus:
2221
2130
)(llcl
slJ
And also:
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Jacobian
Jacobian might be found by directly
differentiating the kinematic equations of
the mechanism for linear velocity,
however there is no 3x1 orientation vectorwhose derivative is rotational velocity.
Thus we get Jacobian using successive
application of:
1
1
1
1
1
1
i
i
ii
ii
ii
i ZR
)( 11
1
1
i
i
i
i
i
ii
ii
i PvRv
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Given a transformation relating joint velocity toCartesian velocity then
Is this matrix invertible? ( Is it non singular)
Singularities
gularitynonJ
ingularityJ
sin:0]det[
s:0]det[
v
J)(
1
)(00 JV
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Singularities
Singularities are categorized into two class:
Workspace boundary singularities:
Occur when the manipulator is fully starched or
folded back on itself.
Workspace interior singularities:
Are away from workspace boundary and are
caused by two or more joint axes lining up.
All manipulators have singularity at boundaries of theirworkspace. In a singular configuration one or more degree offreedom is lost. ( movement is impossible )
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Example
180,0
.00
|)(|)]([
2
221
2221
21
sll
llcl
slJJDET
Workspace boundary singulari ties
2221
2130
)(llcl
slJ
In the earlier example we had:
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Example
.
,
.1
)(
21
12
22
12
21
121
1221112211
122122
221
10
sl
c
sl
c
slc
slslclcl
slcl
sllJ
As the arm stretches out toward 2=0 both
joint rates go to infinity 40
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Static Forces in Manipulators
i
i
n
f force exerted on link i by link i -1
torque exerted on link i by link i-1
Force and moments propagation
To solve for joint
torques in static
equilibrium
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Solve for the joint torques which must be acting
to keep the system in static equi l ibrium.
Static Forces in Manipulators
0111 ii
i
i
i
i
i
i fPnn
01 ii
i
i ff
Summing the force and
setting them equal to zero
Summing the torques aboutthe or igin of frame i
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.
,
.
,
11
1
1
1
1
1
111
1
i
i
i
i
i
ii
ii
i
i
ii
ii
i
i
i
i
i
i
i
i
i
i
i
i
i
fPnRn
fRf
fPnn
ff
Working down from last l ink to the base we
formulate the force moment expressions
Static force propagation
fr om link to link:
Static Forces in Manipulators
Important question:What torques
are needed at the joint to balance
reaction forces and moments acting
on the links?
.
.
i
iT
i
i
i
i
iT
i
i
i
Zf
Zn
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Jacobians in the Force Domain
XF
Work is the dot product of a vector force or torque and a
vector displacement
jX
I t can be written as:
.
.
.
00F
F
FF
T
T
TTTT
J
J
JJ
TT XF
The defini tion of jacobian is
So we have
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Cartesian Transformation ofVelocities and Static Forces
vV
General velocityof a body
General forceof a body
6x 6 transformations map these quantities from one frame
to another.
3 x1 linear velocity
3 x1 angular velocity
N
FF
3 x1 force vector
3 x1 moment vector
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Since two frames are rigidly connected
Cartesian Transformation of Velocitiesand Static Forces
Where the cross product isthe matrix operator
01
i
. 11
11
11
i
iii
iiii
i ZR (5.45)
A
A
AA
B
A
BORGABABA
B
B
BB
w
v
R
PRR
w
v
0
0
00
xy
xz
yz
pp
pppp
P
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A
A
AA
B
A
BORGAB
ABA
B
B
BB v
R
PRRv
0
We use the termvelocity transformation
Description of velocity in terms of A when given the quantities in B
Cartesian Transformation of Velocitiesand Static Forces
A
A
v
A
BB
BvTv
B
B
B
B
A
B
A
BBORG
AA
B
A
A
A
A v
R
RPRv
0
B
B
v
A
BA
A
vTv 47
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A force-moment transformation
With similari ty to Jacobians
Cartesian Transformation of Velocitiesand Static Forces
B
B
B
B
A
B
A
BBORG
A
A
B
A
A
A
A
N
F
RRP
R
N
F 0
B
B
f
A
BA
A FTF
T
v
A
Bf
A
B TT